In this chapter, we study the Banach spaces of the form \(C(X; \mathbb {F})\) for X a compact metrizable Polish space and \(\mathbb {F}\in \{\mathbb {R}, \mathbb {C}\}\) . Recall that this is the space of all continuous functions from X into \(\mathbb {F}\) in which the operations are defined pointwise and the norm is the supremum norm. We call these spaces C-spaces. Again, we organize our presentation around the themes of computable presentability, computable categoricity, and index sets. Before proceeding with these topics, we review some useful classical material on these spaces. We also discuss some work on reducing the signature under consideration.